500 Numbered Cards

Problem Statement

Imagine a deck of 500 cards numbered from 1 to 500. If all the cards are shuffled randomly and you are asked to pick three cards, one at a time, what is the probability of each subsequent card being larger than the previous drawn card?

Solution

To calculate the required probability we need to calculate the share between the number of positive outcomes of the experiment and the total number of possible outcomes. Let's start with the latter: if we pull 3 cards from the deck one after another, there're 500 ways to pull the first card, then 499 ways to pull the second and 498 ways to pull the third.
So, overall we have: $$Total~number~of~outcomes = 500 * 499 * 498$$ Now we need to figure out the number of positive outcomes (i.e. triples in the increasing order). We can calculate the total number of ways to choose 3 cards out of 500 using the formula for combinations: $$C_{500}^{3} = {500! \over 3!~497!} = {500 * 499 * 498 \over 6}$$ Now note that for each particular combination of 3 cards we can have 3 * 2 * 1 = 6 permutations and only one of these permutations will be in a strictly increasing order. It means that the number of combinations (3 out of 500) is exactly the numerator that we need.

The probability we’re looking for is: $$\pmb{P = {C_{500}^{3} \over 500 * 499 * 498} = {1 \over 6}}$$ This answer doesn't really depend on the size of a deck. We could cut the corner and calculate the number of permutations for 3 cards right away, then notice that there's only 1 permutation in the increasing order and get the same result.

Not convinced? Try the simulation below!

Simulation

show / hide simulation code

import random

deck_size = 500
sample_size = 3

nrounds = 10000
positive_outcomes = 0

for i in range(nrounds):
    cards_sample = random.sample(range(1, deck_size + 1), sample_size)
    is_positive = sorted(cards_sample) == cards_sample
    positive_outcomes += is_positive

simulated_probability = round(positive_outcomes / nrounds, 4)
        

Enter the number of experiments
Enter the number of cards in deck
Enter the sample size
	Submit

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